For each given rank \(\hat{r}\)  in \(\lbrace 1,2,\dots ,(p\wedge q)-1\rbrace \) , we derive the estimated coefficient matrix \(\hat{C}\)  using the ridge estimation method [1]}. Denoting the test data as \(\lbrace \widetilde{X},\widetilde{Y}\rbrace \) , we calculate the mean squared prediction error (MSPE) as
\(\mathrm {MSPE}=100\times \frac{\Vert \widetilde{ Y}-\widetilde{ X}\hat{ C}\Vert _{\mathrm {F}}^{2}}{nq}.\) 
